Deck 23: Capacitance and Dielectrics

Question 1

In the circuit shown in the figure, the capacitors are initially uncharged. The switch is first thrown to position A and kept there for a long time. It is then thrown to position B. Let the charges on the capacitors be Q₁, Q₂, and Q₃ and the potential differences across them be V₁, V₂, and V3. Which of the following conditions must be true with the switch in position B?

A) V₁ = V₂ = V₃
B) V₁ + V₂ = V₃
C) V₃ = V₀
D) Q₁ = Q₂ = Q₃
E) Q₁ + Q₂ = Q₃

Answer 1

V₁ + V₂ = V₃

Question 2

Each plate of a parallel-plate air-filled capacitor has an area of 0.0020 m², and the separation of the plates is 0.020 mm. An electric field of 3.9 × 10⁶ V/m is present between the plates. What is the surface charge density on the plates? (ε₀ = 8.85 × 10-12 C²/N ∙ m²)

A) 35 µC/m²
B) 73 µC/m²
C) 17 µC/m²
D) 52 µC/m²
E) 87 µC/m²

Answer 2

35 µC/m²

Question 3

When two or more capacitors are connected in series across a potential difference

A) the potential difference across the combination is the algebraic sum of the potential differences across the individual capacitors.
B) each capacitor carries the same amount of charge.
C) the equivalent capacitance of the combination is less than the capacitance of any of the capacitors.
D) All of the above choices are correct.
E) None of the above choices are correct.

Answer 3

All of the above choices are correct.

Question 4

The charge on the square plates of a parallel-plate capacitor is Q. The potential across the plates is maintained with constant voltage by a battery as they are pulled apart to twice their original separation, which is small compared to the dimensions of the plates. The amount of charge on the plates is now equal to

A) 4Q.
B) 2Q.
C) Q.
D) Q/2.
E) Q/4.

Question 5

When two or more capacitors are connected in parallel across a potential difference

A) the potential difference across each capacitor is the same.
B) each capacitor carries the same amount of charge.
C) the equivalent capacitance of the combination is less than the capacitance of any of the capacitors.
D) All of the above choices are correct.
E) None of the above choices are correct.

Question 6

An ideal air-filled parallel-plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original energy density between the plates was u₀, what is the new energy density?

A) 16u₀
B) 4u₀
C) u₀
D) u₀/4
E) u₀/16

Question 7

The capacitance per unit length of a very long coaxial cable, made of two concentric cylinders, is 50 pF/m. What is the radius of the outer cylinder if the radius of the inner one is 1.0 mm? (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²)

A) 3.0 mm
B) 2.0 mm
C) 4.0 mm
D) 1.0 mm
E) 0.50 mm

Question 8

A charged capacitor stores energy U. Without connecting this capacitor to anything, dielectric having dielectric constant K is now inserted between the plates of the capacitor, completely filling the space between them. How much energy does the capacitor now store?

A) 2KU
B) KU
C) U
D)
E)

Question 9

A parallel-plate capacitor has plates of area 0.40 m2 and plate separation of 0.20 mm. The capacitor is connected across a 9.0-V potential source. (ε0 = 8.85 × 10^-12 C²/N ∙ m²)
(a) What is the magnitude of the electric field between the plates?
(b) What is the capacitance of the capacitor?
(c) What is the magnitude of the charge on each plate of the capacitor?

Question 10

An ideal parallel-plate capacitor consists of a set of two parallel plates of area A separated by a very small distance d. When the capacitor plates carry charges +Q and -Q, the capacitor stores energy U0. If the separation between the plates is doubled, how much electrical energy is stored in the capacitor?

A) 4U₀
B) 2U₀
C) U₀
D) U₀/2
E) U₀/4

Question 11

An ideal parallel-plate capacitor consists of a set of two parallel plates of area A separated by a very small distance d. When this capacitor is connected to a battery that maintains a constant potential difference between the plates, the energy stored in the capacitor is U₀. If the separation between the plates is doubled, how much energy is stored in the capacitor?

A) 4U₀
B) 2U₀
C) U₀
D) U₀/2
E) U₀/4

Question 12

A metal cylinder of radius 2.0 mm is concentric with another metal cylinder of radius 5.0 mm. If the space between the cylinders is filled with air and the length of the cylinders is 50 cm, what is the capacitance of this arrangement? (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²)

A) 33 pF
B) 60 pF
C) 22 pF
D) 30 pF
E) 11 pF

Question 13

The electric field between square the plates of a parallel-plate capacitor has magnitude E. The potential across the plates is maintained with constant voltage by a battery as they are pulled apart to twice their original separation, which is small compared to the dimensions of the plates. The magnitude of the electric field between the plates is now equal to

A) 4E.
B) 2E.
C) E.
D) E/2.
E) E/4.

Question 14

Two capacitors, C₁ and C₂, are connected in series across a source of potential difference. With the potential source still connected, a dielectric is now inserted between the plates of capacitor C₁. What happens to the charge on capacitor C₂?

A) The charge on C₂ increases.
B) The charge on C₂ decreases.
C) The charge on C₂ remains the same.

Question 15

Equal but opposite charges Q are placed on the square plates of an air-filled parallel-plate capacitor. The plates are then pulled apart to twice their original separation, which is small compared to the dimensions of the plates. Which of the following statements about this capacitor are true? (There may be more than one correct choice.)

A) The energy stored in the capacitor has doubled.
B) The energy density in the capacitor has increased.
C) The electric field between the plates has increased.
D) The potential difference across the plates has doubled.
E) The capacitance has doubled.

Question 16

An ideal air-filled parallel-plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original capacitance was C₀, what is the new capacitance?

A) 4C₀
B) 2C₀
C) C₀
D) C₀/2
E) C₀/4

Question 17

The four identical capacitors in the circuit shown in the figure are initially uncharged. Let the charges on the capacitors be Q₁, Q₂, Q₃, and Q₄ and the potential differences across them be V₁, V₂, V₃, and V₄. The switch is thrown first to position A and kept there for a long time. It is then thrown to position B. Which of the following conditions is true with the switch in position B?

A) V₁ = V₂ = V₃ = V₄
B) V₁ = V₀
C) V₁ + V₂ + V₃ + V₄ = V₀
D) Q₁ = 3 Q₂
E) Q₁ = Q₂

Question 18

Two thin-walled concentric conducting spheres of radii 5.0 cm and 10 cm have a potential difference of 100 V between them. (k = 1/4πε0 = 8.99 × 10⁹ N ∙ m²/C²)
(a) What is the capacitance of this combination?
(b) What is the charge carried by each sphere?

Question 19

An ideal air-filled parallel-plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original energy stored in the capacitor was U₀, how much energy does it now store?

A) 4U₀
B) 2U₀
C) U₀
D) U₀/2
E) U₀/4

Question 20

An air-filled parallel-plate capacitor is connected to a battery and allowed to charge up. Now a slab of dielectric material is placed between the plates of the capacitor while the capacitor is still connected to the battery. After this is done, we find that

A) the energy stored in the capacitor had decreased.
B) the voltage across the capacitor had increased.
C) the charge on the capacitor had increased.
D) the charge on the capacitor had not changed.
E) None of these choices are true.

Question 21

Three capacitors are connected as shown in the figure. What is the equivalent capacitance between points a and b?

A) 1.7 µF
B) 4.0 µF
C) 7.1 µF
D) 12 µF
E) 8.0 µF

Question 22

Five capacitors are connected across a potential difference V_ab as shown in the figure. Because of the dielectrics used, each capacitor will break down if the potential across it exceeds 30.0 V. The largest that V_ab can be without damaging any of the capacitors is closest to

A) 6.0 V.
B) 30 V.
C) 64 V.
D) 150 V.
E) 580 V.

Question 23

Four capacitors are connected across a 90-V voltage source as shown in the figure.
(a) What is the charge on the 4.0-μF capacitor?
(b) What is the charge on a 2.0-μF capacitor?
(c) What is the charge on the 3.0-μF capacitor?
(d) What is the potential difference across the 6.0-μF capacitor?

Question 24

The capacitors in the network shown in the figure all have a capacitance of 5.0 µF. What is the equivalent capacitance, Cab, of this capacitor network?

A) 20 µF
B) 3.0 µF
C) 10 µF
D) 5.0 µF
E) 1.0 µF

Question 25

In the circuit shown in the figure, all the capacitors are air-filled. With the switch S open, the 40.0-µF capacitor has an initial charge of 5.00 µC while the other three capacitors are uncharged. The switch is then closed and left closed for a long time. Calculate the initial and final values of the total electrical energy stored in these four capacitors.

Question 26

Three capacitors, with capacitances C₁ = 4.0 μF, C₂ = 3.0 μF, and C₃ = 2.0 μF, are connected to a 12 -V voltage source, as shown in the figure. What is the charge on capacitor C₂?

A) 16 μC
B) 32 μC
C) 2.0 μC
D) 8.0 μC
E) 4.0 μC

Question 27

Three capacitors, of capacitance 5.00 μF, 10.0 μF, and 50.0 μF, are connected in series across a 12.0-V voltage source.
(a) How much charge is stored in the 5.00-μF capacitor?
(b) What is the potential difference across the 10.0-μF capacitor?

Question 28

A 1.0 m long piece of coaxial cable has a wire with a radius of 1.1 mm and a concentric conductor with inner radius 1.3 mm. The area between the cable and the conductor is filled with a dielectric. If the voltage drop across the capacitor is 6000 V when the line charge density is
8)8 μC/m, find the value of the dielectric constant. (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²)

A) 4.4
B) 4.8
C) 5.3
D) 5.7

Question 29

The network shown in the figure is assembled with uncharged capacitors X, Y, and Z, with CX = 7.0 μF, CY = 7.0 μF, and CZ = 6.0 μF, and open switches, S₁ and S₂. A potential difference V_ab = +120 V is applied between points a and b. After the network is assembled, switch S₁ is closed for a long time, but switch S₂ is kept open. Then switch S₁ is opened and switch S₂ is closed. What is the final voltage across capacitor X?

A) 94 V
B) 87 V
C) 79 V
D) 71 V
E) 63 V

Question 30

The capacitive network shown in the figure is assembled with initially uncharged capacitors. A potential difference, V_ab = +100V, is applied across the network. The switch S in the network is initially open but is then closed. Assume that all the capacitances shown are accurate to two significant figures. What is the equivalent capacitance between ab
(a) with the switch S open?
(b) with the switch S closed?

Question 31

A cylindrical capacitor is made of two thin-walled concentric cylinders. The inner cylinder has radius r₁ = 4.0 mm, and the outer one a radius r₂ = 8.0 mm. The common length of the cylinders is L = 150 m. What is the potential energy stored in this capacitor when a potential difference 4.0 V is applied between the inner and outer cylinder? (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²)

A) 9.6 × 10^-8 J
B) 1.3 × 10^-8 J
C) 6.3 × 10^-8 J
D) 0.34 × 10^-8 J
E) 4.6 × 10^-8 J

Question 32

Three capacitors are arranged as shown in the figure. C₁ has a capacitance of 5.0 pF, C₂ has a capacitance of 10.0 pF, and C₃ has a capacitance of 15.0 pF. Find the voltage drop across the entire arrangement if the voltage drop across C₂ is 311 V.

A) 1900 V
B) 1200 V
C) 570 V
D) 520 V

Question 33

An isolated air-filled parallel-plate capacitor that is no longer connected to anything has been charged up to Q = 2.9 nC. The separation between the plates initially is 1.20 mm, and for this separation the capacitance is 31 pF. Calculate the work that must be done to pull the plates apart until their separation becomes 5.30 mm, if the charge on the plates remains constant. (ε₀ = 8.85 × 10^-12 C²/N ∙ m²)

Question 34

A 6.00-μF parallel-plate capacitor has charges of ±40.0 μC on its plates. How much potential energy is stored in this capacitor?

A) 103 μJ
B) 113 μJ
C) 123 μJ
D) 133 μJ
E) 143 μJ

Question 35

Two capacitors of capacitance 6.00 μF and 8.00 μF are connected in parallel. The combination is then connected in series with a 12.0-V voltage source and a 14.0-μF capacitor, as shown in the figure.
(a) What is the equivalent capacitance of this combination?
(b) What is the charge on the 6.00-μF capacitor?
(c) What is the potential difference across the 6.00-μF capacitor?

Question 36

An air-filled capacitor is formed from two long conducting cylindrical shells that are coaxial and have radii of 48 mm and 84 mm. The electric potential of the inner conductor with respect to the outer conductor is -400 V. (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²) The energy stored in a 1.0-m length of this capacitor is closest to

A) 8.0 μJ.
B) 5.7 μJ.
C) 11 μJ.
D) 16 μJ.
E) 22 μJ.

Question 37

The capacitive network shown in the figure is assembled with initially uncharged capacitors. A potential difference, V_ab = +100V, is applied across the network. The switch S in the network is kept open. Assume that all the capacitances shown are accurate to two significant figures. What is potential difference V_cd across the open switch S?

A) 0 V
B) 40 V
C) 50 V
D) 60 V
E) 70 V

Question 38

A charge of 2.00 μC flows onto the plates of a capacitor when it is connected to a 12.0-V potential source. What is the minimum amount of work that must be done in charging this capacitor?

A) 6.00 µJ
B) 24.0 µJ
C) 12.0 µJ
D) 144 µJ
E) 576 µJ

Question 39

An air-filled capacitor is formed from two long conducting cylindrical shells that are coaxial
And have radii of 13 mm and 85 mm. The electric potential of the inner conductor with respect to the outer conductor is -600 V. (k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²) The maximum energy density of the capacitor is closest to

A) 2.7 × 10^-3 J/m³.
B) 1.3 × 10^-3 J/m³.
C) 6.7 × 10^-4 J/m³.
D) 3.4 × 10^-3 J/m³.
E) 1.7 × 10^-4 J/m³.

Question 40

A 1.0-μF and a 2.0-μF capacitor are connected in series across a 3.0-V voltage source.
(a) What is the charge on the 1.0-μF capacitor?
(b) What is the voltage across the 2.0-μF capacitor?

Question 41

A parallel-plate capacitor, with air between the plates, is connected across a voltage source. This source establishes a potential difference between the plates by placing charge of magnitude 4.15 × 10^-6 C on each plate. The space between the plates is then filled with a dielectric material, with a dielectric constant of 7.74. What must the magnitude of the charge on each capacitor plate now be, to produce the same potential difference between the plates as before?

Question 42

A 6.0-μF air-filled capacitor is connected across a 100-V voltage source. After the source fully charges the capacitor, the capacitor is immersed in transformer oil (of dielectric constant 4.5). How much ADDITIONAL charge flows from the voltage source, which remained connected during the process?

A) 1.2 mC
B) 1.5 mC
C) 1.7 mC
D) 2.1 mC
E) 2.5 mC

Question 43

A 15-μF air-filled capacitor is connected to a 50-V voltage source and becomes fully charged. The voltage source is then removed and a slab of dielectric that completely fills the space between the plates is inserted. The dielectric has a dielectric constant of 5.0.
(a) What is the capacitance of the capacitor after the slab has been inserted?
(b) What is the potential difference across the plates of the capacitor after the slab has been inserted?

Question 44

A parallel-plate capacitor consists of two parallel, square plates that have dimensions 1.0 cm by 1.0 cm. If the plates are separated by 1.0 mm, and the space between them is filled with teflon, what is the capacitance of this capacitor? (The dielectric constant for teflon is 2.1, and ε₀ = 8.85 × 10^-12 C²/N ∙ m².)

A) 1.9 pF
B) 0.44 pF
C) 2.1 pF
D) 0.89 pF

Question 45

A 1.0 μF capacitor has a potential difference of 6.0 V applied across its plates. If the potential difference across its plates is increased to 8.0 V, how much ADDITIONAL energy does the capacitor store?

A) 14 μJ
B) 28 μJ
C) 2.0 μJ
D) 4.0 μJ

Question 46

Each plate of an air-filled parallel-plate air capacitor has an area of 0.0040 m², and the separation of the plates is 0.080 mm. An electric field of 5.3 × 10⁶ V/m is present between the plates. What is the energy density between the plates? (ε0 = 8.85 × 10^-12 C²/N ∙ m²)

A) 124 J/m³
B) 84 J/m³
C) 170 J/m³
D) 210 J/m³
E) 250 J/m³

Question 47

A parallel-plate capacitor with plate separation of 1.0 cm has square plates, each with an area of 6.0 × 10^-2 m². What is the capacitance of this capacitor if a dielectric material with a dielectric constant of 2.4 is placed between the plates, completely filling them? (ε0 = 8.85 × 10^-12 C²/N ∙ m²)

A) 15 × 10^-12 F
B) 15 × 10^-14 F
C) 64 × 10^-14 F
D) 1.3 × 10^-12 F
E) 1.3 × 10^-10 F

Question 48

Two square air-filled parallel plates that are initially uncharged are separated by 1.2 mm, and each of them has an area of 190 mm². How much charge must be transferred from one plate to the other if 1.1 nJ of energy are to be stored in the plates? (ε₀₀ = 8.85 × 10^-12 C²/N ∙ m²)

A) 56 pC
B) 39 pC
C) 78 pC
D) 3.5 µC

Question 49

A parallel-plate capacitor has a capacitance of 10 mF and is charged with a 20-V power supply. The power supply is then removed and a dielectric material of dielectric constant 4.0 is used to fill the space between the plates. What is the voltage now across the capacitor?

A) 80 V
B) 20 V
C) 10 V
D) 5.0 V
E) 2.5 V

Question 50

The capacitive network shown in the figure is assembled with initially uncharged capacitors. A potential difference, V_ab = +100V, is applied across the network. The switch S in the network is kept open. Assume that all the capacitances shown are accurate to two significant figures. What is the total energy stored in the seven capacitors?

A) 48 mJ
B) 72 mJ
C) 96 mJ
D) 120 mJ
E) 144 mJ

Question 51

A parallel-plate capacitor has a capacitance of 10 mF and charged with a 20-V power supply. The power supply is then removed and a dielectric material of dielectric constant 4.0 is used to fill the space between the plates. How much energy is now stored by the capacitor?

A) 250 mJ
B) 125 mJ
C) 500 mJ
D) 62.5 mJ
E) 1200 mJ

Question 52

An air-filled capacitor stores a potential energy of 6.00 mJ due to its charge. It is accidentally filled with water in such a way as not to discharge its plates. How much energy does it continue to store after it is filled? (The dielectric constant for water is 78 and for air it is 1.0006.)

A) 0.077 mJ
B) 468 mJ
C) 0.040 mJ
D) 6.00 mJ